Environmental Engineering Reference
In-Depth Information
1.2.3
Statistical Weight of a State and Distributions of Particles in Gases
Above we used subscript
i
to refer to one particle state, whereas below we consider
a general case where
i
characterizes a set of degenerate states. We introduce the
statistical weight
g
i
of a state that is a number of degenerate states
i
. For example,
a diatomic molecule in a rotational state with the rotational quantum number
J
has the statistical weight
g
i
1, equal to the number of momentum projec-
tions on the molecular axis. Including accounting for the statistical weight, formula
(1.41) takes the form
D
2
J
C
Cg
j
exp
T
,
ε
j
n
j
D
(1.42)
where
C
is the normalization factor and the subscript
j
refers to a group of states.
In particular, this formula gives the relation between the number densities
N
0
and
N
j
of particles in the ground and excited states, respectively:
g
0
exp
T
,
N
0
g
j
ε
j
N
j
D
(1.43)
where
j
is the excitation energy and
g
0
and
g
j
are the statistical weights of the
ground and excited states.
We now determine the statistical weight of states in a continuous spectrum. The
wave function of a free particle with momentum
p
x
moving along the
x
-axis is
given by exp(
ip
x
x
/
ε
) to within an arbitrary factor if the particle is moving in the
positive direction and by exp(
„
ip
x
x
/
„
) if the particle is moving in the negative
10
27
erg s is the Planck constant
h
divided
direction. (The quantity
„D
1.054
by 2
.) Suppose that the particle is in a potential well with infinitely high walls.
Theparticlecanmovefreelyintheregion0
π
L
and the wave function at the
walls goes to zero. To construct a wave function that corresponds to free motion
inside the well and goes to zero at the walls, we superpose the basic free-particle
solutions, so
<
x
<
ψ
D
C
1
exp(
ip
x
x
/
„
)
C
C
2
exp(
ip
x
x
/
„
). From the boundary condi-
tion
ψ
(0)
D
0itfollowsthat
ψ
D
C
sin(
p
x
x
/
„
), and from the second boundary
condition
n
,where
n
is an integer. This procedure
thus yields the allowed quantum energies for a particle moving in a rectangular
well with infinitely high walls.
From this it follows that the number of states for a particle with momentum in
the range from
p
x
to
p
x
ψ
(
L
)
D
0weobtain
p
x
L
/
„D
π
),wherewetakeinto
account the two directions of the particle momentum. For a spatial interval
dx
,the
number of particle states is
C
dp
x
is given by
dn
D
Ldp
x
/(2
π
„
dp
x
dx
2
dn
D
.
π
„
Generalizing to the three-dimensional case, we obtain
dp
x
dx
2
dp
y
dy
2
dp
z
dz
2
d
p
d
r
(2
dn
D
D
)
3
.
(1.44)
π
„
π
„
π
„
π
„